Modules over Projective Schemes

Definition 1. Let S be a graded ring, set X = ProjS and letM a graded S-module. We define a sheaf of modulesM ̃ on X as follows. For each p ∈ ProjS we have the local ring S(p) and the S(p)module M(p) (GRM,Definition 4). Let Γ(U,M ̃) be the set of all functions s : U −→ ∐p∈U M(p) with s(p) ∈M(p) for each p, which are locally fractions. That is, for every p ∈ U there is an open neighborhood p ∈ V ⊆ U and m ∈ M,f ∈ S of the same degree, such that for every q ∈ V we have f / ∈ q and s(q) = m/f ∈ M(q). It is easy to check that M ̃ is a sheaf of modules with the obvious restriction maps and the action (r · s)(p) = r(p) ·m(p). If φ : M −→ N is a morphism of graded S-modules then for every p ∈ ProjS we have a canonical morphism of S(p)-modules φ(p) : M(p) −→ N(p) (GRM,Definition 4) and we define a morphism of sheaves of modules